The structural index in gravity and magnetic interpretation: Errors, uses, and abuses

The structural index (SI) is based on the concept of Euler homogeneity, a description of scaling behavior. It has found wide use in potential-field depth estimation and is a constant integer for simple sources with single singularities (points, lines, thin-bed faults, sheet edges, infinite contacts). For these cases, the SI is identical to the index of a simple power-law field fall-off with distance. The simple Euler formulation is only strictly correct for such simple sources and integer SI values. The widespread use of the simple Euler method on more complex structures, using fractional SI values is likely to produce misleading results because the SI is no longer a constant for any given source. We examine a recently published example that used an arbitrary SI to estimate depth to the base of the crust for Africa and produced misleading results. Extension to more complex sources such as tabular bodies or thick steps requires one of several more generalized approaches, which recognize all variables with spatial dimensions (including source size parameters) and may make use of negative SI values, address omitted variable bias or use an explicit multiple-source formulation. An alternative approach using homogeneity via differential similarity transforms is probably the best way forward. An error in the literature is corrected: the gravity SI for a finite step is -1, but it requires a more generalized formulation. We develop a new terminology, fractional SI, s, which is permitted to take fractional values and makes no pretense to honor concepts of homogeneity.